$hide=mobile

[Booklet] Vietnamese Mathematical Competitions 2017

The Vietnam Mathematical Olympiad 2017 for high school students took place in all cities of Vietnam on 05-06th January, 2017. After that competition, a two-day IMO Team selection test was organized. The top 48 students and one student who got a Silver medal at IMO 2016 took two tests on 25th and 26th of March.

Based on the TST result, the Vietnamese team for the 58th International Mathematical Olympiad 2017 in Rio de Janeiro, Brazil were selected: Phan Nhat Duy, Le Quang Dung, Nguyen Canh Hoang, Hoang Huu Quoc Huy, Pham Nam Khanh, Do Van Quyet

Dr. Le Anh Vinh (The Vietnam Institute of Educational Sciences - Hanoi) and Dr. Le Ba Khanh Trinh (Vietnam National University - Ho Chi Minh city) were appointed as the Leader and the Deputy Leader of the Vietnamese IMO team 2017. The team is also accompanied by Mr. Nguyen Khac Minh (Ministry of Education and Training). This booklet, includes problems and solutions of VMO and TST this year, which was edited by Le Phuc Lu (FPT Software Ho Chi Minh City), Nguyen Van Quy (Hanoi University of Science). We also say thanks to the trainers help us in completion the booklet: Dr. Tran Nam Dung, Vo Quoc Ba Can, Tran Quang Hung, Nguyen Van Linh, Nguyen Huy Tung, Luong Van Khai.

Vietnam National Mathematical Olympiad 2017

  1. Given $a\in\mathbb{R}$ and a sequence $(u_n)$ defined by \[u_1=a,\quad u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}},\,\forall n\in\mathbb{N}^*.\] a) Prove that $(u_n)$ is convergent sequence when $a=5$ and find the limit of the sequence in that case.
    b) Find all $a$ such that the sequence $(u_n)$ is exist and is convergent. 
  2. Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases}\]
  3. Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. $AH$ intersects $(O)$ at $D$ ($D\ne A$).
    a) Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN\perp OH$.
    b) The lines $DE$, $DF$ intersect $(O)$ at $P$, $Q$ respectively ($P\ne D$, $Q\ne D$). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ($R\ne A$, $S\ne A$). Prove that $BP$, $CQ$, $RS$ are concurrent. 
  4. Given an integer $n>1$ and a $n\times n$ grid $ABCD$ containing $n^2$ unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called symmetry if each unit square has center on diagonal $AC$ is colored by gray and every couple of unit squares which are symmetry by $AC$ should be both colred by black or white. In each gray square, they label a number $0$, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called $k$-balance (with $k\in\mathbb{Z}^+$) if it satisfies the following requirements
    • Each pair of unit squares which are symmetry by $AC$ are labelled with the same integer from the closed interval $[-k,k]$.
    • If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct.
    a) For $n=5$, find the minimum value of $k$ such that there is a $k$-balance label for the following grid
    b) Let $n=2017$. Find the least value of $k$ such that there is always a $k$-balance label for a symmetry coloring. 
  5. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying relation $$f(xf(y)-f(x))=2f(x)+xy,\,\forall x,y \in \mathbb{R}$$
  6. Prove that
    a) $\displaystyle \sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ $(\bmod 2017^2)$.
    b) $\displaystyle \sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ $(\bmod 2017^2)$ 
  7. Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ($E\ne A, F\ne A$).
    a) Let $K$ be the intersection of $BE$ and $CF$. Prove that $AK,BC,OG$ are concurrent.
    b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$) of $(I)$. $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$. Assume that $MN$ intersects $(O)$ at $P$ nad $Q$. Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$, the circumcircle $(GPQ)$ always passes through two fixed points.

Vietnam Team Selection Test 2017

  1. There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide. 
  2. For each positive integer $n$, set $x_n=\binom{2n}{n}$.
    a) Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$.
    b) Find all positive integer $h>1$ such that there exists positive integers $N$, $T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.
  3. Triangle $ABC$ with incircle $(I)$ touches the sides $AB$, $BC$, $AC$ at $F$, $D$, $E$, res. $I_b$, $I_c$ are $B$- and $C$- excenters of $ABC$. $P$, $Q$ are midpoints of $I_bE$, $I_cF$. $(PAC)\cap AB=\{ A, R\}$, $(QAB)\cap AC=\{ A,S\}$.
    a) Prove that $PR$, $QS$, $AI$ are concurrent.
    b) $DE$, $DF$ cut $I_bI_c$ at $K$, $J$, res. $EJ\cap FK=\{ M\}$. $PE$, $QF$ cut $(PAC)$, $(QAB)$ at $X$, $Y$ res. Prove that $BY$, $CX$, $AM$ are concurrent.
  4. Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order.
    a) Prove that $SR$ passes through the midpoint of $KP$.
    b) Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point. 
  5. Given $2017$ positive real numbers $a_1,a_2,\dots ,a_{2017}$. For each $n>2017$, set $$a_n=\max\{a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.$$Prove that there exists a positive integer $m\leq 2017$ and a positive integer $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for every $n>N$. 
  6. For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called beautiful if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$).
    a) For $n=6$, point out a beautiful permutation.
    b) Prove that there exists a beautiful permutation for every $n$.

Post a Comment


$hide=mobile

$hide=mobile

$hide=mobile

$show=post$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0

Kỷ Yếu$cl=violet$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0$hide=mobile

Journals$cl=green$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0$hide=mobile

Name

Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,3,Amsterdam,5,Ấn Độ,2,An Giang,23,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,52,Bắc Giang,50,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,48,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,38,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,14,Bình Định,45,Bình Dương,23,Bình Phước,38,Bình Thuận,34,Birch,1,Booklet,11,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,1,BxMO,13,Cà Mau,14,Cần Thơ,14,Canada,40,Cao Bằng,7,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,351,Chu Tuấn Anh,1,Chuyên Đề,124,Chuyên Sư Phạm,31,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,618,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,56,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,1766,Đề Thi JMO,1,Điện Biên,8,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,4,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,49,Đồng Tháp,52,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,17,ELMO,19,EMC,9,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,26,Gauss,1,GDTX,3,Geometry,12,Gia Lai,26,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,29,Hà Nội,232,Hà Tĩnh,72,Hà Trung Kiên,1,Hải Dương,50,Hải Phòng,42,Hàn Quốc,5,Hậu Giang,4,Hậu Lộc,1,Hilbert,1,Hình Học,33,HKUST,7,Hòa Bình,13,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,101,HSG 11,91,HSG 12,585,HSG 9,425,HSG Cấp Trường,78,HSG Quốc Gia,106,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,33,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,56,IMT,1,India,45,Inequality,13,InMC,1,International,315,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,17,KHTN,54,Kiên Giang,64,Kim Liên,1,Kon Tum,18,Korea,5,Kvant,2,Kỷ Yếu,42,Lai Châu,4,Lâm Đồng,33,Lạng Sơn,21,Langlands,1,Lào Cai,17,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,1,Leibniz,1,Long An,42,Lớp 10,10,Lớp 10 Chuyên,455,Lớp 10 Không Chuyên,229,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lý Thánh Tông,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,11,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,10,MYM,227,MYTS,4,Nam Định,33,Nam Phi,1,Nam Trung Bộ,1,National,249,Nesbitt,1,Newton,4,Nghệ An,52,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,8,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,42,Ninh Thuận,15,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,99,Olympic 10/3,5,Olympic 11,92,Olympic 12,30,Olympic 24/3,7,Olympic 27/4,20,Olympic 30/4,69,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,12,Olympic Toán,304,Olympic Toán Sơ Cấp,3,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,30,Phú Yên,29,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,45,Putnam,25,Quảng Bình,44,Quảng Nam,32,Quảng Ngãi,34,Quảng Ninh,43,Quảng Trị,27,Quỹ Tích,1,Riemann,1,RMM,12,RMO,24,Romania,36,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,69,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia,7,Scholze,1,Serbia,17,Sharygin,24,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,27,Sóc Trăng,28,Sơn La,12,Spain,8,Star Education,5,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,14,Tập San,6,Tây Ban Nha,1,Tây Ninh,29,Thạch Hà,1,Thái Bình,39,Thái Nguyên,49,Thái Vân,2,Thanh Hóa,62,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,7,Thừa Thiên Huế,36,Tiền Giang,19,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,126,Trà Vinh,6,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,34,Trại Hè Hùng Vương,25,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,9,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,14,Trường Đông,19,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,56,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Sinh 10,680,Tuyển Tập,44,Tuymaada,4,Undergraduate,67,USA,44,USAJMO,10,USATST,7,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,2,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,27,Vĩnh Long,21,Vĩnh Phúc,64,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,47,VNTST,22,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,18,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
ltr
item
MOlympiad: [Booklet] Vietnamese Mathematical Competitions 2017
[Booklet] Vietnamese Mathematical Competitions 2017
https://2.bp.blogspot.com/-tEyc78C24dU/WieoUgASZkI/AAAAAAAAA8E/pAHYuDdREAElPonXKcujOrbcZFt5WcrQQCLcBGAs/s200/vmo-2017.png
https://2.bp.blogspot.com/-tEyc78C24dU/WieoUgASZkI/AAAAAAAAA8E/pAHYuDdREAElPonXKcujOrbcZFt5WcrQQCLcBGAs/s72-c/vmo-2017.png
MOlympiad
https://www.molympiad.net/2017/12/vietnamese-mathematical-competitions-2017.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2017/12/vietnamese-mathematical-competitions-2017.html
true
2506595080985176441
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy